PHYSICAL REVIEW D 103, 063022 (2021)

INTEGRAL x-ray constraints on sub-GeV dark matter

Marco Cirelli ,1,* Nicolao Fornengo ,2 Bradley J. Kavanagh ,3 and Elena Pinetti 1,2 1Laboratoire de Physique Th´eorique et Hautes Energies (LPTHE), UMR 7589 CNRS & Sorbonne University, 4 Place Jussieu, Paris F-75252, France 2Dipartimento di Fisica, Universit`a di Torino & INFN, Sezione di Torino, via P. Giuria 1, Torino I-10125, Italy 3Instituto de Física de Cantabria, (IFCA, UC-CSIC), Av. de Los Castros s/n, Santander 39005, Spain

(Received 2 September 2020; accepted 4 February 2021; published 17 March 2021)

Light dark matter (DM), defined here as having a mass between 1 MeVand about 1 GeV, is an interesting possibility both theoretically and phenomenologically, at one of the frontiers of current progress in the field of DM searches. Its indirect detection via gamma rays is challenged by the scarcity of experiments in the MeV– GeV region. We look therefore at lower-energy x-ray data from the INTEGRAL telescope, and compare them with the predicted DM flux. We derive bounds which are competitive with existing ones from other techniques. Crucially, we include the contribution from inverse Compton scattering on galactic radiation fields and the cosmic microwave background, which leads to much stronger constraints than in previous studies for DM masses above 20 MeV.

DOI: 10.1103/PhysRevD.103.063022

I. INTRODUCTION NEWS-G [21], DARKSIDE [22],EDELWEISS[23],andin the future LBECA [24]),possibly exploitingthe productionof The possibility that dark matter (DM) consists of a light detectable signals via DM-electron scattering [25–28],the particle has gained increasing attention recently (for def- Migdal effect1 [15,36,37] or the interactions of DM with the initeness we intend here a mass between a few MeV and Sun or with cosmic rays [38–50]. Efforts are underway as well about a GeV, as we discuss below). Searches for DM have to develop new detection strategies [51,52], including the use long been dominated by the paradigm of (heavier) weakly of semiconductors [53–55] (such as in the SENSEI [56] and interacting massive particles (WIMPs) [1,2], but with no DAMIC [57,58] experiments), superconductors [59,60], convincing WIMP signal observed so far in direct detection superfluid helium [61–66], evaporating helium [67],2D [3], indirect detection [4–6] or collider searches [7,8], the materials such as arrays of nanotubes and graphene attention is turning to lighter (or heavier) candidates. Light [68,69], 1D materials such as superconducting nanowires DM is indeed in some sense a new frontier for searches, [70], chemical bond breaking [71], production of color-center requiring new analysis strategies and experimental tech- defects in crystals [72], polar materials (i.e., crystals with niques to achieve sensitivity [9]. easily excitable polar atomic bonds [73,74]), paleodetectors In direct detection (DD), most current experiments lose [75,76], magnetic bubble chambers [77], Casimir forces sensitivity for DM masses below ∼1 GeV. This is because the between nucleons [78], aromatic targets [79],molecular standard method of detection relies on detecting the small gas excitations [80], or diamonds [81]. Such a long list amounts of energy deposited by DM via nuclear recoils, illustrates the strong interest in the DD community for this which becomes ineffective for DM much lighter than a typical regime, and bears well for the future. nucleus. However, many significant efforts are underway to On the side of collider searches, the situation has some explore the sub-GeV regime. This includes extending the resemblance with the DD case. The current experiments sensitivity of “traditional” nuclear recoil detectors to ultralow and flagship colliders (essentially the LHC) are not suitable energy thresholds [10,11] (XENON-10 [12], XENON-1T [13],LUX[14,15], CRESST [16–19], SUPER-CDMS [20], 1In the standard DD method, the ionization signal is typically produced as the nucleus struck by the DM particle recoils and hits *[email protected] in turn other nuclei, freeing their electrons. This requires a certain non-negligible energy deposition. The Migdal effect [29–35] Published by the American Physical Society under the terms of refers instead to the fact that even a slowly recoiling or just the Creative Commons Attribution 4.0 International license. shaken nucleus can lose one or some of its electrons, and Further distribution of this work must maintain attribution to therefore produce a signal. Standard ionization is an in-medium the author(s) and the published article’s title, journal citation, process, while the Migdal effect can take place even for the and DOI. Funded by SCOAP3. scattering of an isolated atom.

2470-0010=2021=103(6)=063022(18) 063022-1 Published by the American Physical Society CIRELLI, FORNENGO, KAVANAGH, and PINETTI PHYS. REV. D 103, 063022 (2021)

2 for exploring the low mass region: the staple signature of value E0 to a final value E ≈ 4γ E0 upon scattering off an DM production is the missing energy corresponding to the electron with relativistic factor γ ¼ Ee=me.Hence,a1GeV DM mass, which in the sub-GeV case is swamped in the electron will produce a ∼1.5 keV x ray when scattering off −4 experimental background. The strategy is therefore to turn the CMB (E0 ≈ 10 eV). By the same token, a mildly to the search for associated states, i.e., particles that belong relativistic MeV electron will produce a ∼0.15 keV x ray “ ” to a new dark sector, more extended than just the DM when scattering off UV starlight (E0 ≈ 10 eV). These con- particle. In particular, if sub-GeV DM is to be produced by siderations roughly define our range of interest for DM ≃ 1 → 1 4 thermal freeze-out in the early Universe, this requires the masses: mDM MeV GeV. So our goal is to explore existence of new light force mediators (“dark photons”). whether there are cases in which x-ray observations can These mediators provide a link between the dark and impose constraints on sub-GeV DM that would otherwise visible sectors and are therefore subject to active searches. fall below the sensitivity of the more conventional gamma- Indeed many theory analyses [82–87] and experimental ray searches. To this end, we focus on data from the projects [88–91] pursue this direction. For a detailed and INTEGRAL x-ray satellite [121]. recent review, see, e.g., Ref. [92], where additional refer- The electrons and positrons produced by the annihilation ences can be found. of light DM propagate in the Galaxy. The dominant energy In indirect detection (ID), one typically searches for the loss processes for these particles at low energy are Standard Model particles (charged particles, such as elec- bremsstrahlung, ionization and Coulomb scattering. trons and positrons, or neutral ones, such as gamma rays These therefore reduce the power that is effectively injected and neutrinos) produced in the annihilation of DM in the into ICS. However these processes are relevant where gas is Galaxy, with energies at or just below the DM mass. present, i.e., essentially in the Galactic center and within the Concerning charged particles, the problem is that solar disk. We will therefore focus on observational windows at activity holds back sub-GeV charged cosmic rays and 2 mid-to-high Galactic latitudes, where indeed ICS (together therefore we have no access to them. Concerning gamma with synchrotron emission) is a relevant mechanism of rays, the sensitivity of the most powerful of the recent photon production. telescopes, Fermi-LAT, stops at about 100 MeV, so that From the theoretical point of view, the case for sub-GeV typically DM masses only as low as about 1 GeV can be DM is arguably not as strong as the traditional case for probed. At much lower energies, below a few MeV, one has WIMPs. On the other hand, considering the lack of evidence ∼1 competitive data from INTEGRAL. But in between and for a DM in the typical WIMP mass range, it is definitely 250 MeV, only relatively old data from COMPTEL are worth leaving no stone unturned [122]. From the phenom- available and no current competitive experiment exists. enological perspective, light (scalar) DM has been invoked as Indeed, a number of authors have discussed proposals to fill a viable possibility [123–125] and as an explanation this so-called MeV gap in a useful way for DM searches [126–129] of the 511 KeV line from the center of the galaxy – [94 106]. Alternatively, low energy neutrinos have been [130]. From the point of view of theoretical motivations, well considered, but they are found not to be very competitive founded models include for instance the strongly inter- with respect to photons (e.g., the projected sensitivity of acting massive particles (SIMP) scenarios [131–135],the 20 years of run with the future Hyper-Kamiokande detector WIMP-less idea [136], forbidden dark matter [137], axinos is weaker than the existing bound that we will discuss [138,139],neutrinomassgeneration[128,140],andeven below, and the new ones we will derive) [107,108]. certain supersymmetry configurations [141–143]: all these Another possibility for ID of such light DM (which is the constructions naturally include or even predict sub-GeV DM one we will entertain here) is to look at a range of energies 3 particles. One may also add to the list asymmetric DM [144]: much lower than that of the DM mass. The basic idea is the in the standard lore it naturally predicts DM with a mass of a following: the electrons and positrons produced in the few GeVand no annihilation signals, but actually the mass can Galactic halo by the annihilations of DM particles with a ≃ 1 ≲ 1 be lighter [145,146] and residual or late annihilations can mass m GeV have naturally an energy E GeV; they occur and give rise to an interesting phenomenology undergo inverse Compton scattering (ICS) on the low energy [145,147]. Freeze-in DM production [148] is also often cited photons of the ambient bath (the cosmic microwave back- in connection to sub-GeV DM, not necessarily because ground (CMB), infrared light and starlight) and produce x freeze-in points to the sub-GeV range but because it provides rays, which can be searched for in x-ray surveys. Indeed, the a viable DM production mechanism. ICS process increases the photon energy from the initial low More broadly, studies addressing the motivations, the phenomenology and the constraints (notably from cosmol- 2An exception to this point is the use of data from the Voyager ogy) of MeV–GeV DM include [149–165]. spacecraft, which is making measurements outside of the helio- sphere [93]. We will comment on the corresponding constraints later. 4Note that we are not interested in keV DM, that can, e.g., 3For former applications of the same idea to heavy, WIMP- produce x rays by direct annihilation or decay. That is a whole like, DM see for instance [109–118]. other set of searches, e.g., for keV sterile neutrino DM [119,120].

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The rest of this paper is organized as follows. In Sec. II subdominant outside of the gas-dense region of the Galactic we review the computation of x-rays from DM annihila- Center that we do not consider, as mentioned above. tions, via the ICS process. In Sec. III we present the x-ray The differential flux of the final state radiation or measurements by INTEGRAL that we employ. In Sec. IV radiative decays photons from the annihilations of a DM we present the results, in particular deriving constraints in particle of mass mDM is computed via the standard the usual plane of annihilation cross section versus mass of expression (see, e.g., Ref. [169]): the DM particle, and we briefly compare with other existing 2 f constraints. In Sec. V we summarize and conclude. dΦ γ 1 ρ dN γ FSR r⊙ ⊙ θ σ FSR Ω ¼ 2 4π Jð Þh vif ; dEγd mDM dEγ II. X RAYS FROM DM ANNIHILATIONS Z ds ρðrðs; θÞÞ 2 JðθÞ¼ ; ð4Þ In this section we briefly review the basic formalism for l:o:s: r⊙ ρ⊙ (soft γ ray and) hard x-ray production from DM annihi- 3 lations, essentially via final state radiation and ICS emis- where r⊙ ≃ 8.33 kpc and ρ⊙ ¼ 0.3 GeV=cm are the sion. For all details we refer the reader to Refs. [166,167], conventional values for the distance of the Sun from the whose notation we mostly follow. Two important dif- Galactic Center (GC) and the local DM density. Here, Eγ ferences however apply. First, Refs. [166,167] do not deal denotes the photon energy, dΩ is the solid angle, r is the with DM masses smaller than 5 GeV, so that we cannot distance from the GC in spherical coordinates, in turn straightforwardly use the tools developed there. Second, in expressed in terms of s, the coordinate that runs along the Refs. [166,167] a detailed treatment of the energy-loss and line of sight (s ¼ 0 corresponds to the Earth position), and diffusion process is employed. Since, as mentioned above, 2 2 2 the angle θ: r ¼ s þ r⊙ − 2sr⊙ cos θ. We use natural we are mostly interested in higher latitude signals where units c ¼ ℏ ¼ 1 throughout the paper. The normalized J ICS energy losses dominate, we adopt a simplified treat- factor corresponds to the integration along the line of sight ment (described below). of the square of the galactic DM density profile ρðrÞ, for Let us consider a direction of observation from Earth that which we adopt a standard Navarro-Frenk-White profile θ is identified by the angle (the aperture between the [166,170] direction of observation and the axis connecting the Earth to the Galactic Center), or, equivalently, by the latitude and −2 l ρ ρ rs 1 r longitude pair ðb; Þ. At each point along this direction, NFWðrÞ¼ s þ with DM particles annihilate, contributing to the photon signal, r rs ρ 0 184 3 24 42 which we collect by integrating all the contributions along s ¼ . GeV=cm ;rs ¼ . kpc: ð5Þ the line of sight. Since we focus on DM lighter than a few GeV, we consider only three annihilation channels: The annihilation cross section in each of the final states f of – σ Eqs. (1) (3) is denoted h vif. → þ − DM DM e e ; ð1Þ For the spectrum of FSR photons we adopt the following expressions, which we derive from the analysis of [171] þ − DM DM → μ μ ; ð2Þ and reproduce here for the convenience of the reader:

− → πþπ − DM DM ; ð3Þ dNlþl α 1 ν FSRγ A þ Rð Þ− 2B ν ¼ 2 ln Rð Þ ; ð6Þ dEγ πβð3 − β Þm 1 − RðνÞ which are kinematically open whenever mDM >mi (with DM i ¼ e, μ, π) and that we study one at a time. The pion channel is representative of a hadronic DM channel. We do where not consider the annihilation into a pair of neutral pions, since in this case the (boosted to the DM frame) γ rays do 1 β2 3 − β2 A ð þ Þð Þ − 2 3 − β2 2ν not reach down to the energies covered by INTEGRAL. ¼ ν ð Þþ ; ð7Þ For each channel, the total photon flux is given by the sum of two contributions: the emission from the charged 3 − β2 particles in the final state (from final state radiation, FSR, B ¼ ð1 − νÞþν ; ð8Þ and, whenever relevant, other radiative decays, Rad) and ν the photons produced via ICS by DM-produced energetic μ ν β2 e. Also, another contribution to the total photon flux is where l ¼ e, and we have defined: p¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEγ=mDM, ¼ þ 1–4μ2 μ 2 ν 1 − 4μ2 1 − ν given by the in-flight annihilation of DM-produced e with with ¼ ml=ð mDMÞ and Rð Þ¼ =ð Þ. ambient e− from the gas [168]. This process is however For the pion [171]

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− dNπþπ 2α ν 1 − ν 1 β2 1 ν FSRγ − ν þ − 1 þ Rð Þ ¼ πβ 2 ν Rð Þþ 2ν ln 1 − ν ; ð9Þ dEγ mDM β Rð Þ with the same definitions as above with ml → mπ. − − The muon can undergo the radiative decay μ → e ν¯eνμγ, and this channel can be a relevant source of soft photons. For the photon spectrum we adopt the parametrization of Refs. [172,173] which in turn are derived from Ref. [174]. In the muon rest frame the expression we adopt is therefore μ α 1 − 1 − dNRad γ ð xÞ 2 x ¼ 12ð3 − 2xð1 − xÞ Þ log þ xð1 − xÞð46 − 55xÞ − 102 ; ð10Þ dEγ 36πEγ r Eμ¼mμ

2 max where x ¼ 2Eγ=mμ, r ¼ðme=mμÞ and the maximal photon energy is Eγ ¼ mμð1 − rÞ=2 ≃ 52.8 MeV. For muons in → μþμ− flight, Eq. (10) is boosted to the frame where the muon has energy Eμ ¼ mDM. For the process DM DM ,a multiplicity factor of 2 needs to be applied, since a pair of muons is produced for each annihilation event. − − Charged pions can also produce photons radiatively through the process π → l ν¯lγ with l ¼ e, μ. Also in this case we adopt the parametrization of Ref. [172], which has been derived from Ref. [175]. In the pion rest frame the expression we adopt is then

π α dNRad γ ½fðxÞþgðxÞ ¼ 2 2 2 ; ð11Þ dEγ 24πmπfπ r − 1 x − 1 rx Eπ ¼mπ ð Þ ð Þ

2 where x ¼ 2Eγ=mπ, r ¼ðml=mπÞ , fπ ¼ 92.2 MeV is the pion decay constant and

2 4 2 2 2 2 fðxÞ¼ðr þ x − 1Þ½mπx ðF þ F Þðr − rx þ r − 2ðx − 1Þ Þ ffiffiffi A V p 2 − 12 2fπmπrðx − 1Þx ðFAðr − 2x þ 1ÞþxFVÞ 2 2 −24fπrðx − 1Þð4rðx − 1Þþðx − 2Þ Þ; ffiffiffi p 2 r 2 gðxÞ¼12 2fπrðx − 1Þ log ½mπx ðF ðx − 2rÞ − xF Þ 1 − x A V ffiffiffi p 2 2 þ 2fπð2r − 2rx − x þ 2x − 2Þ; ð12Þ

2 2 where the axial and vector form factors are FA ¼ 0.0119 and FVðq Þ¼FVð0Þð1 þ aq Þ with FVð0Þ¼0.0254, a ¼ 0.10, q2 ¼ð1 − xÞ [172,176]. When the pion decays into an on shell muon, the radiative decay of the muon is again a source of low energy photons. Following Ref. [172], the total radiative charged pion spectrum in the pion rest frame can be phrased as

π π μ dN X dN dN Rad tot γ π → lν Rad γ π → μν Rad γ ¼ BRð lÞ þ BRð μÞ ; ð13Þ dEγ l μ dEγ dEγ Eπ ¼mπ ¼e; Eπ¼mπ Eμ¼E⋆

2 2 where E⋆ ¼ðmπ þ mμÞ=ð2mπÞ is the muon energy in the in the derivation on the DM annihilation cross section pion rest frame. For the process DM DM → πþπ−, where bounds in Sec. IV. the pion is in flight, Eq. (13) is boosted to the frame where The differential flux of the inverse Compton scattering Eπ ¼ mDM and a multiplicity factor of 2 is applied, since photons received at Earth is written in terms of the also in this case a pair of pions is produced for each emissivities jðEγ; x⃗Þ of all cells located along the line of annihilation event. As pointed out in Ref. [172], the muon sight at position x⃗: Z radiative decay provides a quite relevant soft-photon Φ ⃗ l d ICγ 1 jðEγ; xðs; b; ÞÞ production channel. Both in the muon and in the pion ¼ ds : ð14Þ dEγdΩ Eγ 4π annihilation channel, the Rad photon flux arising from the l:o:s: muon radiative decay is dominant over the FSR emission Here the spherical symmetry of the system around the for DM masses up to a few GeV. This will be explicitly seen center of the Galaxy is broken by the distribution of the

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10–13 In the Gal Plane: Above the Gal Center: 10–14 (R,z) = (8,0) kpc (R,z) = (0,4) kpc

10–15 ] 10–16

–17

[GeV/sec 10 input e± energy: (R,z)=(8.33,0) kpc 10–11 b 100 MeV 1 GeV 10 GeV 10–18 10–12 CMB 10–19 10 MeV –13 10 10–20 IR –21 10–14 10 Ioniz Ioniz

1 MeV Energy loss coefficient

ICS power [1/sec] –22 Brems Brems –15 10 10 Opt ICS ICS –23 10 Synchrotron Synchrotron 10–16 –24 –6 –5 –4 –3 –2 –1 2 3 4 10 10 10 10 10 10 10 1101010 10 1 10 102 103 104 1 10 102 103 104 photon energy E [MeV] e± energy [MeV] e± energy [MeV]

FIG. 1. Left: ICS power as a function of the emitted photon energy, for the listed input e energies. In the 10 GeV case, for illustration we show the contributions of the three components of the photon bath: CMB, infrared light and optical light. Right: energy losses for electrons and positrons in the energy regime of interest. The first subpanel refers to the situation in the Galactic plane, in a spot similar to the location of the solar system: one sees that the e lose most of their energy via interactions with the gas (ionization and bremsstrahlung). The second subpanel is for a point well outside the plane, on the vertical of the Galactic Center: one sees that the ICS emission is dominant down to ∼40 MeV. Lines of sight that avoid the Galactic plane (i.e., at high latitude) are therefore preferred for our purposes. ambient light, which mostly lies in the galactic disk. Hence and references therein) but, for reference, we plot the total we use the 3D notation x⃗instead of r, and we need the power P in the energy regime of interest in Fig. 1, left. In l θ σ ðb; Þ coordinates instead of the single angle . These the above equation, IC is the Klein-Nishina cross section. quantities are related by the usual expression cos θ ¼ The local e spectral number density we are concerned cos b cos l. It will also be convenient to work in cylindrical with here is the result of injection due to DM annihilation, coordinates ðR; zÞ. The emissivity is obtained as a con- plus the subsequent diffusion of the e in the local Galactic volution of the density of the emitting medium with the environment, subject to energy losses by several processes power that it radiates. In this case therefore (see below). We adopt a simplified treatment by neglecting Z diffusion, i.e., we assume that electrons and positrons m DM dne scatter off the ambient photons in the same location where jðEγ; x⃗Þ¼2 dE P ðEγ;E ; x⃗Þ ðE ; x⃗Þ; ð15Þ e IC e e me dEe these e were produced by annihilation events of DM particles. This is sometimes referred to as the “on the spot” where Ee denotes the electron total energy and dne =dEe is approximation. While this is certainly a drastic approxi- the local electron (or positron) spectral number density (the mation, we verified a posteriori that it is justified in the overall factor of 2 takes into accountP the equal populations regimes of our interest: the timescales of diffusion over P Pi of electrons and positrons). ¼ i IC is the differential lengths comparable to the angular bins of the INTEGRAL power emitted into photons due to ICS radiative processes data are longer than the loss timescales, except for lines of Z sight towards high latitudes, which however contribute Pi ⃗ ϵ ϵ ⃗ σ ϵ little to the signal. In this regime, the quantity can be simply ðEγ;Ee; xÞ¼Eγ d nið ; xÞ ICð ;Eγ;EeÞ; ð16Þ IC expressed as (see, e.g., [115]) Z where the sum on i runs over the different components of dn 1 mDM e ðE ; x⃗Þ¼ dE˜ Q ðE˜ ; x⃗Þ; the photon bath, expressed by their photon number den- dE e b ðE ; x⃗Þ e e e sities n (ϵ is the photon energy before the scattering): e tot e Ee i σ ρ ⃗ 2 CMB, dust-rescattered infrared light (IR) and optical star- ˜ ⃗ h vi ðxÞ dNe with QeðEe; xÞ¼ 2 ˜ : ð17Þ light (SL). For the last two components, we use the mDM dEe interstellar radiation field (ISRF) maps extracted from ⃗ ≡ − dE [177],asin[115,166,167]. We do not detail here further Here btotðE; xÞ dt ¼ bCoulþioniz þ bbrem þ bsyn þ bICS the computations (we refer the reader to [115,166,167,178] is the energy loss function, which takes into account all

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FIG. 2. Example photon spectra from sub-GeV DM that illustrate our main points. Left: the hard x-ray and soft γ-ray spectrum produced by a 150 MeV DM particle annihilating into μþμ−. We show the different components in color and the total flux in thick black. The spectrum cuts off before reaching Fermi’s data (taken from [182] and reported here just for reference, as they are not the focus of our work), but produces a signal in x rays that can be constrained by INTEGRAL (taken from [121], Fig. 7). The FSR (blue dashed) and Rad (blue dot- dashed) contributions yield signals that pass well below the x-ray data. However, the inclusion of the DM-induced ICS contribution on the different components of the Galactic ambient light (StarLight (SL, green dotted), dust-reprocessed infrared light (IR, brown dotted) ) and the CMB (not visible in the plot), leads to a flux which is orders of magnitude larger, thus producing stronger constraints. Right: the same for a 10 MeV DM particle annihilating into eþe−. In this case the limit is instead driven by the FSR contribution because the DM ICS contributions fall to too low energy for INTEGRAL. In these illustrations, the signals are computed over the jbj < 15°; jlj < 30° region of interest: in our analysis we actually use smaller regions of interest, removing low latitudes (see Sec. III).

the energy loss processes that the e suffer in the local where ϱ ¼ me=mμ, ξ ¼ 2Ee=mμ and the maximal electron max 2 2 þ − Galactic environment in which they are injected. These are energy is Ee ¼ðmμ þ meÞ=ð2mμÞ.Fortheπ π case, due due (in order of importance for increasing e energy: to the decay chain π → μ → e, we boost the electron Coulomb interactions and ionization, Bremsstrahlung, syn- spectrum from muon decay of Eq. (18) to the rest frame of 2 2 chrotron emission, and inverse Compton scattering). We plot the pion [where the muon has energy Eμ ¼ðmπ þ mμÞ= the energy loss function in Fig. 1 (right) for illustration and ð2mπÞ] and then boost the ensuing distribution to the DM we refer the reader to [167] for more details. It is important to annihilation frame (where the pion has energy Eπ ¼ mDM). stress that these quantities are subject to astrophysical The expression of the boost from a frame where the particle a uncertainties, stemming, e.g., from the uncertainties in (produced by its parent A) has energy E0 and momentum p0 to the gas distribution, in the ISRF and in the intensity of the a frame where the parent particle has energy EA [180] reads galactic magnetic field. We will discuss the impact of the Z such uncertainties on our signal at the end of this section. 1 E0 1 ˜ dN max dN QeðEe; x⃗Þ is the injection term of electrons and positrons ¼ 0 0 ; ð19Þ dE 2βγ 0 p dE ˜ Emin from DM annihilations: its integral over Ee corresponds essentially to the number of electrons generated with an −2 1=2 where γ ¼ EA=mA and β ¼ð1 − γ Þ are the Lorentz energy larger than Ee at position x⃗. We cut the integral in 20 factors for the boost, and E0 ¼ γðE βpÞ and the the horizontal direction at R ¼ Rgal ¼ kpc, where Rgal is maxjmin the presumed radial extension of the Milky Way, and in the signs þ (−) refers to max (min), respectively. When a is 0 vertical direction at z ¼ 4 kpc, assuming this to be the size of produced as monochromatic, i.e., dN=dE ¼ δðE − E⋆Þ,the the magnetic halo that keeps the e confined. above expression reduces to the typical box spectrum The last ingredient needed, the e spectrum from DM dN=dE ¼ 1=ð2βγp⋆Þ for γðE⋆ − βp⋆Þ ≤ E ≤ γðE⋆ þ βp⋆Þ. þ − annihilations in the different cases, is rather straightfor- We remind the reader here that our π π case is just wardly computed. For the eþe− channel it consists simply intended as one possible representative case of DM annihilations into light quarks. A more detailed analysis in a monochromatic line of e with Ee ¼ mDM. For the μþμ− case, we boost to the DM annihilation frame (where would require using the spectra computed in [181] that consider the annihilation into light quark pairs and the the muon has energy Eμ ¼ mDM) the electron spectrum from muon decay obtained in the muon rest frame [179]: subsequent production of many light hadronic resonances besides pions. That would require choosing a specific pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ→eνν¯ 2 2 model for the annihilation diagram (e.g., via a light dNe 4 ξ − 4ϱ ¼ ½ξð3 − 2ξÞþϱ2ð3ξ − 4Þ; ð18Þ mediator with arbitrary but defined couplings to quarks). dEe mμ In order to keep our analysis as simple and as model

063022-6 INTEGRAL X-RAY CONSTRAINTS ON SUB-GeV DARK MATTER PHYS. REV. D 103, 063022 (2021) independent as possible, we stick to the case of direct affect the energy losses by ICS and of course the ICS signal annihilation into charged pions only, deferring a more emission. We vary by a factor of 2 overall the intensity of complete analysis to future work. the ISRF in the Galaxy to mimic this error. Finally, the With all the ingredients above, we are therefore in a galactic magnetic field also carries significant uncertainties, position to compute the full spectrum of photons from DM which impact the energy losses by synchrotron. We adopt annihilation. The final step consists in integrating the the different magnetic field configurations discussed in contributions in Eqs. (4) and (14) over the selected region [167]. The effect however is very limited, since the of observation: synchrotron radiation losses are always subdominant in Z Z our regime of interest (see Fig. 1). Figure 3 illustrates the l dΦ γ bmax max dΦ γ dΦ γ impact of these uncertainties on the spectra. DM ¼ db dl cos b FSR þ IC : dEγ b l dEγdΩ dEγdΩ min min III. INTEGRAL X-RAY DATA AND ANALYSIS ð20Þ We use the data from the INTEGRAL/SPI x-ray spec- We can then compare the total flux with the data in the same trometer,asreportedin[121], which follows previous work in region, in order to derive constraints on dark matter. Two [183,184]. The data were collected in the period 2003–2009, examples are presented in Fig. 2, illustrating the main corresponding to a significant total exposure of about 108 points of our analysis. seconds, and cover a range in energy between 20 KeV and a few MeV.They are provided both in the form of a spectrum of A. Astrophysical uncertainties the total diffuse flux in a rectangular region of observation centered around the GC (jbj < 15°; jljj < 30°, Figs. 6 and 7 As mentioned above, the computed spectra are subject to in [121]) and in the form of an angular flux in latitude and astrophysical uncertainties of different sorts, which we longitude bins (Figs. 4 and 5 in [121]). In Fig. 2,for detail here. First of all, the DM distribution in the Galaxy is illustration, we showed the former set of data. In the uncertain. Choosing a different profile (with respect to the following, however, we will use the latter set of data, from NFW profile specified above) affects all the different which we cut out the Galactic plane (GP). This is based on two components: FSR, rad, and the three ICS components. reasons, both of which make the GP less attractive from an To compute the constraints, we will therefore also adopt a analysis perspective. First, the GP is obviously bright in x rays cored profile and a peaked NFW one (characterized by a due to many astrophysical sources (dust emission, IC emission 1 26 slope r . towards the GC). Second, the gas density in the from cosmic rays, etc.), which have no connection to dark Galaxy has significant uncertainties. This affects the energy matter but represent a significant source of background. In our losses by Coulomb, ionization, and bremsstrahlung, there- analysis, in order to adopt a conservative approach in the fore in turn affecting the spectrum of the emitting e.We derivation of the DM bounds, we will not attempt to model and will vary by a factor of 2 the overall gas density in the subtract the galactic x-ray emission and we therefore choose to Galaxy. Third, the ISRF also carries uncertainties, which stay away from the most contaminated region. Second, the e emissivity in the GP is dominated by scattering processes on gas (Coulomb interactions, ionization, and bremsstrahlung), due to its high density there. This complexity of the GP induces uncertainties in the predictions of the photon fluxes, which can be avoided by looking at higher latitudes. Therefore we choose to focus on relatively gas-poorer regions in which the ICS emissivity is the most relevant process. This also represents a conservative choice. The data adopted in our analysis are shown in Fig. 4 (adapted from Fig. 5 of [121]). They are divided into five energy bands (27–49, 49–90, 100–200, 200–600, and 600– 1800 keV), each featuring 21 bins in latitude (15 in the case of the fifth band). The longitude window is −23.1° < l < 23.1° for the first four bands, and −60° < l < 60° for the fifth one. For each DM annihilation channel, we compute the total FIG. 3. Impact of the astrophysical uncertainties on the same photon flux from DM annihilation in each energy band and photon spectra of Fig. 2 (left panel). The shaded bands corre- spond to the cumulative effect of the uncertainty on the DM per each latitude/longitude bin as discussed in Sec. II.In profile, on the gas density, on the ISRF and on the galactic Fig. 4 we show as black dots one example of the predicted magnetic field. Each colored band correspond to the different photon flux which refers to a 150 MeV DM particle − components (blue shade for FSR and Rad, green shade for ICS on annihilating into μþμ . We see that the DM flux increases SL, salmon shade for ICS on IR light). approaching the central latitude bins, as expected, because

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FIG. 4. INTEGRAL data, compared to the maximal predicted DM flux: in each of the five energy bands, we show the data from INTEGRAL (colored points) as well as the predicted DM flux for 150 MeV DM annihilating into μþμ− (black points). The shades cover the low-latitude points that we do not use. In each panel the annihilation cross section is set at its limit for this DM mass within that specific energy band: i.e., it maximizes the flux without exceeding the data in the band by more than allowed by our constraint procedure. of the larger DM density towards the center of the Galaxy. fact that the energy losses are large for electrons in the If the emission were simply due to FSR, the angular profile plane as a consequence of the high gas density, and would strictly follow the DM density profile squared. This therefore their ICS emissivity is suppressed. As discussed is actually not the case here, since the ICS signal, which is above, we remove (“mask”) the three central latitude bins in dominant in the configuration considered here, is molded order to exclude most of the signal from the GP.5 by the spatial dependences of the e energy losses and of the density of the target radiation fields. For instance, the 5For the first four energy bands, the three central bins cover the DM flux in the very central latitude bin, which corresponds interval −3.9°

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.

FIG. 5. Our conservative constraints on sub-GeV DM from INTEGRAL data (solid thick lines), compared to other existing bounds: from Voyager 1 e data (dashed green and blue lines, from Boudaud et al. [93]), from a compilation of x-ray data (dot-dashed green line, from Essig et al. [188]) and from the CMB assuming s-wave (dotted green and blue lines in the lower portion of the plot, from Slatyer [189] and Lopez-Honorez et al. [190])orp-wave annihilation (dotted green and blue lines in the upper portion of the plot, from 2 220 100 2 Diamanti et al. [191] and Liu et al. [192]; these bounds are rescaled up by a factor ðv=vref Þ ¼ð = Þ since they are provided in the 100 ≃ 220 literature for vref ¼ km=s while we consider v km=s in the Milky Way). For each probe, we use the color code specified in the legend: green for the DM DM → eþe− annihilation channel, blue for DM DM → μþμ− and magenta for DM DM → πþπ−. When results on a channel are not present in the literature, the corresponding color is missing. For instance, there are no bounds from these probes on the πþπ− channel besides ours.

We then proceed to derive constraints on dark matter in than the measured value. This means that bins where the two different ways: we first derive conservative bounds by predicted DM flux is smaller that the observed one are not not including any astrophysical galactic x-ray emission; we considered incompatible with the observations. The DM then derive more optimistic limits by adopting a model for flux starts to introduce tension only when it exceeds the data the astrophysical background and adding a DM component points, and this tension progressively increases with the DM on top of it. We discuss the two strategies in turn in the annihilation cross section. For each DM mass (raster scan), following. this is equivalent to having a number of unconstrained (non- negative) nuisance parameters for the background in each 2 A. Conservative constraints bin, which makes our statistic equivalent to a Δχ , distrib- uted as a χ2 with 1 degree of freedom (see also [185] for a From the comparison between the predicted DM flux similar approach). We then impose a bound on hσvi when Φ ϕ DMγ and the measured flux we derive constraints on the 2 6 χ> ¼ 4, which corresponds to a 2σ bound. DM annihilation cross section by requiring that the former does not exceed the latter by more than an appropriate B. Optimistic constraints amount. More precisely, we proceed as follows. We define a test statistic Reference [121] also provides templates of the different astrophysical emission processes, computed within GALPROP, X X 2 [187] as an attempt to explain the measured flux. Taken ðMax½ðΦ γ ðhσviÞ − ϕ Þ; 0Þ χ2 DM ;i i collectively, these astrophysical emissions constitute a back- > ¼ σ2 ; ð21Þ bands i∈fb binsg i ground flux ϕB. Including this in our analysis clearly leads to stronger constraints, since the room for exotic components is where the first sum runs over the five INTEGRAL energy reduced. To derive DM bounds, we adopt the following bands and the second over the latitude bins (except the three (standard) procedure. We multiply the astrophysical 2 central ones) and σi is the error bar on the ith data point. χ> 2 corresponds to computing a global “effective” χ that 6For a detailed discussion of similar test statistics, see, e.g., includes only the data bins where the DM flux is higher Ref. [186], especially Secs. 2.5 and 3.7.

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FIG. 6. Dissection of the constraints. First three panels: bound from each INTEGRAL energy band, for the three annihilation channels consi- dered. Bottom-right panel: comparison between the bounds obtained using Rad, FSR, or ICS only (but combining the five energy bands). background flux by an overall, energy-independent7 normali- by combining the data from all angular and energy bins and 2 zation factor NB. We consider the χ statistic for the total photon flux originating from both FSR and ICS processes. Figures 6 and 7 elucidate the impact on the X X 2 Φ γ σv N ϕ − ϕ bounds coming from the various elements of the analysis. χ2 ð DM ;iðh iÞ þ B B iÞ ¼ σ2 : ð22Þ The first three panels in Fig. 6 show the constraints bands i∈fbbinsg i imposed by each energy band separately. They are obtained with the same χ2 criterion defined in Eq. (21) independ- σ > We identify the pair ðNB;0; h vi0Þ that minimizes the statistic, ently in each energy band. We can see that the dominant corresponding to the value χ0. Then, for each DM mass, we constraint often (although not always) comes from the 49– scan the values of ðN;hσviÞ and we impose a constraint on 90 keV band. The highest energy band (600–1800 keV) 2 2 2 hσvi when Δχ ¼ χ − χ0 ¼ 4,foranyNB. almost always provides the weakest bound. The relative strength of the bounds depends on a number of factors, IV. RESULTS AND DISCUSSION including statistical fluctuations in the data and the size of the errors bars in the different energy bands, but most In this section we present the constraints on dark matter notably from the position (in energy) of the ICS peak annihilation that we derive following the procedure dis- contribution relative to the INTEGRAL data. Let us stress cussed in the previous section. Figure 5 is our main result, that the global constraints (in Fig. 5) are not just the lower as it shows the overall (conservative) bounds on the envelope of the curves in these panels: they are in fact annihilation cross section hσvi as a function of the DM computed using all the data points in all energy bands mass for each of the three annihilation channels, obtained simultaneously. In order to understand the origin of the constraints shown in the first three panels, the last (bottom right) panel of 7We explored the possibility of having a different normaliza- tion of the background in the different energy bands. This Fig. 6 shows the bounds obtained by considering only the assumption turned out to only mildly affect the overall constraints FSR, the Rad or the ICS contributions to the photon flux. in the DM parameter space. As anticipated above, the ICS constraint is dominant when

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FIG. 7. Impact of the methodology and the uncertainties on the constraints. Left: constraints with and without astrophysical background. Right: variation of the bounds due to the astrophysical uncertainties. See text for details. present. This is because for larger DM masses the ICS flux heliosphere, using different propagation assumptions (we is significantly higher than the FSR and Rad one in the report the bounds of their model B, characterized by weak energy range of INTEGRAL data (see the illustration in reacceleration). Their constraints intertwine with ours over Fig. 2 left). The ICS flux shifts to lower energies when the the mass range under consideration, being stronger in the DM mass decreases, since only lower-energy eþe− can be mass range 7–100 MeV and weaker otherwise. produced: this can be appreciated from the ICS power The CMB constraints derived in [189] are the most depicted in Fig. 1 (left). As the mass becomes smaller than stringent across the whole mass range (they are given in about 30 MeV (which can, in fact, occur only for the eþe− [189] for the eþe− channel and in the earlier study [190] for channel), the ICS contribution peaks at energies lower than the μþμ− channel, in the mass range of interest).9 However, the band covered by INTEGRAL and therefore becomes they hold under the assumption that DM annihilation is ineffective, thus leaving the FSR emission to dominate the speed independent (s wave). If the DM annihilation is constraint. instead p wave, i.e., hσvi ∝ v2, they weaken considerably. In Fig. 7, the left panel shows the bounds obtained with This can be understood qualitatively with the following the two procedures discussed in Sec. III: conservative and argument (see [191] and the discussion in [188] for a more optimistic. One sees that the optimistic bounds are more precise assessment). The CMB constrains the energy stringent, as expected, by about half an order of magnitude. injection from DM annihilations at high redshift (at the The right panel shows the cumulative impact of the time of recombination or somewhat later). For p-wave astrophysical uncertainties, as discussed in Sec. II.One annihilating DM, such injection was suppressed since the sees that they span up to two orders of magnitude. DM was very cold (slow) back then. In the galactic halo, at Figure 5 also shows the comparison with other existing present times, DM particles move faster, as an effect of the constraints. Essig et al. [188] have derived bounds using a gravitational collapse that formed large scale structures, compilation of x-ray and soft γ-ray data from HEAO-1, and therefore annihilate more efficiently. In other words, a INTEGRAL, COMPTEL, EGRET, and Fermi. Among the large value for the annihilation cross section at present day annihilation channels that we study, they consider only is allowed as it corresponds to a much smaller value and eþe−. They do not include the ICS and they use hence a limited effect at the time of the CMB. Instead the INTEGRAL data in the region jbj < 15°; jlj < 30° rather bounds obtained in our analysis, and the other bounds that than the latitude bins that we use (from which we exclude we report, are sensitive only to DM annihilation at the the Galactic plane). Their bound is comparable with ours at present time and therefore are independent of the s-wave/ small masses, becoming stronger in the mass range p-wave assumption if we assume, as usually done, a 5–40 MeV due to the inclusion of COMPTEL data, then constant DM speed in the galactic halo. If instead we ≳ 50 becoming weaker for mDM MeV when ICS emission introduce a radial dependence of the DM speed, the p-wave sets in.8 bounds are affected. We have estimated that they depart − Boudaud el al. [93] have derived constraints on the eþe from the s-wave ones by a factor Oð40%Þ, for typical and μþμ− channel using low energy measurements by assumptions on the DM speed and density profile in the Voyager 1 of the e cosmic ray flux outside of the Galaxy.

8Laha et al. [193], in v1 on the arXiv, also present a result in 9Additional bounds, somehow weaker that the CMB ones, can agreement with Essig et al. [188], while in v2 the bound is no be obtained using only the DM effect on the temperature of the longer present. intergalactic medium [194].

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Fermi constraints as computed by the collaboration (e.g., probe sub-GeV annihilating DM. At the same time, [195]) are not provided for DM masses below a few GeV, upcoming full-sky data from the eROSITA x-ray telescope therefore we do not report them here. (which will cover the range 0.3–10 keV) [200] will be A few other studies directly connected with our sensitive to the ICS emission of sub-GeV DM and will analysis have appeared in the literature in the past. therefore likely allow us to improve the reach of the References [172,196] have introduced tools for computing technique we used in this paper. signals like those in which we are interested, which however we do not employ. Reference [196] also derives ACKNOWLEDGMENTS bounds, but focuses mostly on synchrotron emission in dwarf galaxies from multi-GeV DM. Reference [197] M. C. and B. J. K. acknowledge the hospitality of the applies our approach, but considers ICS on the CMB only, Institut d’Astrophysique de Paris (IAP) where part of this 1 focuses on mDM > GeV and uses only a partial meas- work was done. N. F. and E. P. acknowledge the hospitality urement from the Draco dSph galaxy. Hence there is no of the Laboratoire de Physique Th´eorique et Hautes overlap of our results with these analyses. Energies (LPTHE), a research unit of CNRS and Sorbonne University, where this work was started in V. CONCLUSIONS 2016 and resumed in 2020. We thank Ennio Salvioni for We have derived and presented constraints on dark comments and we acknowledge very useful discussions matter in the mass range 1 MeV to 5 GeV, comparing with John Beacom concerning in-flight annihilation and x-ray emission from the annihilation of such light DM with with Adam Coogan concerning the radiative decays. We data from the INTEGRAL telescope. Our constraints (see also acknowledge the following organizations for support: Fig. 5) are comparable with previous results derived using European Research Council (ERC) under the EU Seventh x-ray data and using e data from Voyager 1. However, Framework Programme (Grant No. FP7/2007-2013)/ERC —“ ” the bounds we present here are the strongest to date on the Starting Grant (Agreement No. 278234 NEWDARK “ ” present-day annihilation of dark matter for masses in the project); CNRS 80|Prime grant ( DAMEFER project) (M. range 150 MeV to 1.5 GeV. CMB bounds remain stronger C.); Departments of Excellence grant awarded by the over the whole mass range, but they do rest on the Italian Ministry of Education, University and Research assumption that the DM annihilation cross section at the (MIUR) (N. F. and E. P.); Research grant of the Universit`a time of recombination is the same as the present-day one. Italo-Francese, under Bando Vinci 2020 (E. P.); Research When this is not the case, the CMB bounds largely relax. grant The Dark Universe: A Synergic Multimessenger The strength of our constraints is due in large part to the Approach, Grant No. 2017X7X85K funded by the inclusion of ICS emission, produced by the upscattering of Italian Ministry of Education, University and Research ambient photons by electrons and positrons produced by (MIUR); Research grant The Anisotropic Dark Universe, dark matter annihilation. The energy of these ICS photons Grant No. CSTO161409, funded by Compagnia di is typically a few orders of magnitude lower than the DM Sanpaolo and University of Torino; Research grant mass, allowing us to use data from INTEGRAL to help TASP (Theoretical Astroparticle Physics) funded by plug the “MeV gap” and produce novel constraints on sub- Istituto Nazionale di Fisica Nucleare (INFN) (N. F. and GeV DM. E. P.); Spanish Agencia Estatal de Investigación (AEI, In the near future, data from eASTROGAM (300 keV– MICIU) for the support to the Unidad de Excelencia 3 GeV) [198] or AMEGO (200 keV–10 GeV) [199] will María de Maeztu Instituto de Física de Cantabria, Ref. hopefully cover the “MeV gap” allowing us to directly No. MDM-2017-0765 (B. J. K.).

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